If $x : y = 3 : 4$, then $(2x + 3y) : (3y - 2x)$ would be equal to

### Concept & Direct Value Substitution When given a ratio of variables $x : y = a : b$, you can substitute $x = ak$ and $y = bk$ into homogeneous algebraic expressions to find their ratio. Because the degree of the numerator and denominator match, the constant $k$ will cancel out, meaning you can plug in the raw ratio values directly. $$ \text{Substitute } x = 3, y = 4 $$ ### Step-by-Step Solution 1. From the given proportion $x : y = 3 : 4$, we can assign placeholder values $x = 3$ and $y = 4$ to evaluate the homogeneous expression. 2. The expression we need to evaluate is $\frac{2x + 3y}{3y - 2x}$. 3. Substitute $x = 3$ and $y = 4$ into the first part of the expression (numerator): $2(3) + 3(4) = 6 + 12 = 18$ 4. Substitute $x = 3$ and $y = 4$ into the second part of the expression (denominator): $3(4) - 2(3) = 12 - 6 = 6$ 5. Form the final ratio using the calculated parts: $18 : 6$ 6. Simplify the ratio by dividing both sides by their greatest common divisor, which is 6: $3 : 1$ ### Exam Strategy & Shortcut Because every term in the expression $(2x + 3y) : (3y - 2x)$ has a degree of 1, directly plugging in $x=3$ and $y=4$ is entirely mathematically sound and skips the tedious step of carrying a proportionality constant $k$ through the arithmetic. ### Common Pitfall A common oversight is performing the subtraction in the second part incorrectly by not strictly following the $3y - 2x$ order, often swapping it to $2x - 3y$ and resulting in a negative ratio. Always double-check the order of operations in the denominator. ### Final Answer Therefore, the correct answer is **$3 : 1$**.